| blob | deb0b5c3f7f619295d2a4426fcaf70574073438e |
1 /*
2 * Copyright 2008-2009 Katholieke Universiteit Leuven
3 * Copyright 2012-2013 Ecole Normale Superieure
4 * Copyright 2014-2015 INRIA Rocquencourt
5 * Copyright 2016 Sven Verdoolaege
6 *
7 * Use of this software is governed by the MIT license
8 *
9 * Written by Sven Verdoolaege, K.U.Leuven, Departement
10 * Computerwetenschappen, Celestijnenlaan 200A, B-3001 Leuven, Belgium
11 * and Ecole Normale Superieure, 45 rue d’Ulm, 75230 Paris, France
12 * and Inria Paris - Rocquencourt, Domaine de Voluceau - Rocquencourt,
13 * B.P. 105 - 78153 Le Chesnay, France
14 */
16 #include <isl_ctx_private.h>
17 #include <isl_map_private.h>
19 #include <isl/map.h>
20 #include <isl_seq.h>
22 #include <isl_space_private.h>
23 #include <isl_mat_private.h>
24 #include <isl_vec_private.h>
26 #include <bset_to_bmap.c>
27 #include <bset_from_bmap.c>
28 #include <set_to_map.c>
29 #include <set_from_map.c>
32 {
36 }
39 {
44 }
45 }
49 {
51 isl_int gcd;
64 }
67 }
75 }
77 }
85 }
88 }
95 }
99 }
103 {
106 }
108 /* Reduce the coefficient of the variable at position "pos"
109 * in integer division "div", such that it lies in the half-open
110 * interval (1/2,1/2], extracting any excess value from this integer division.
111 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
112 * corresponds to the constant term.
113 *
114 * That is, the integer division is of the form
115 *
116 * floor((... + (c * d + r) * x_pos + ...)/d)
117 *
118 * with -d < 2 * r <= d.
119 * Replace it by
120 *
121 * floor((... + r * x_pos + ...)/d) + c * x_pos
122 *
123 * If 2 * ((c * d + r) % d) <= d, then c = floor((c * d + r)/d).
124 * Otherwise, c = floor((c * d + r)/d) + 1.
125 *
126 * This is the same normalization that is performed by isl_aff_floor.
127 */
130 {
131 isl_int shift;
146 }
148 /* Does the coefficient of the variable at position "pos"
149 * in integer division "div" need to be reduced?
150 * That is, does it lie outside the half-open interval (1/2,1/2]?
151 * The coefficient c/d lies outside this interval if abs(2 * c) >= d and
152 * 2 * c != d.
153 */
156 {
157 isl_bool r;
169 }
171 /* Reduce the coefficients (including the constant term) of
172 * integer division "div", if needed.
173 * In particular, make sure all coefficients lie in
174 * the half-open interval (1/2,1/2].
175 */
178 {
183 isl_bool reduce;
193 }
196 }
198 /* Reduce the coefficients (including the constant term) of
199 * the known integer divisions, if needed
200 * In particular, make sure all coefficients lie in
201 * the half-open interval (1/2,1/2].
202 */
205 {
219 }
222 }
224 /* Remove any common factor in numerator and denominator of the div expression,
225 * not taking into account the constant term.
226 * That is, if the div is of the form
227 *
228 * floor((a + m f(x))/(m d))
229 *
230 * then replace it by
231 *
232 * floor((floor(a/m) + f(x))/d)
233 *
234 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
235 * and can therefore not influence the result of the floor.
236 */
238 {
254 }
256 /* Remove any common factor in numerator and denominator of a div expression,
257 * not taking into account the constant term.
258 * That is, look for any div of the form
259 *
260 * floor((a + m f(x))/(m d))
261 *
262 * and replace it by
263 *
264 * floor((floor(a/m) + f(x))/d)
265 *
266 * The difference {a/m}/d in the argument satisfies 0 <= {a/m}/d < 1/d
267 * and can therefore not influence the result of the floor.
268 */
271 {
283 }
285 /* Assumes divs have been ordered if keep_divs is set.
286 */
289 {
307 }
318 }
327 /* We need to be careful about circular definitions,
328 * so for now we just remove the definition of div k
329 * if the equality contains any divs.
330 * If keep_divs is set, then the divs have been ordered
331 * and we can keep the definition as long as the result
332 * is still ordered.
333 */
340 }
341 }
343 /* Assumes divs have been ordered if keep_divs is set.
344 */
347 {
355 }
357 /* Check if elimination of div "div" using equality "eq" would not
358 * result in a div depending on a later div.
359 */
362 {
377 }
380 }
382 /* Eliminate divs based on equalities
383 */
386 {
401 isl_bool ok;
417 }
418 }
422 }
424 /* Eliminate divs based on inequalities
425 */
428 {
458 }
460 }
462 /* Does the equality constraint at position "eq" in "bmap" involve
463 * any local variables in the range [first, first + n)
464 * that are not marked as having an explicit representation?
465 */
468 {
477 isl_bool unknown;
486 }
489 }
491 /* The last local variable involved in the equality constraint
492 * at position "eq" in "bmap" is the local variable at position "div".
493 * It can therefore be used to extract an explicit representation
494 * for that variable.
495 * Do so unless the local variable already has an explicit representation or
496 * the explicit representation would involve any other local variables
497 * that in turn do not have an explicit representation.
498 * An equality constraint involving local variables without an explicit
499 * representation can be used in isl_basic_map_drop_redundant_divs
500 * to separate out an independent local variable. Introducing
501 * an explicit representation here would block this transformation,
502 * while the partial explicit representation in itself is not very useful.
503 * Set *progress if anything is changed.
504 *
505 * The equality constraint is of the form
506 *
507 * f(x) + n e >= 0
508 *
509 * with n a positive number. The explicit representation derived from
510 * this constraint is
511 *
512 * floor((-f(x))/n)
513 */
516 {
518 isl_bool involves;
541 }
545 {
568 }
577 progress);
584 }
591 }
594 }
598 {
600 progress));
601 }
605 {
611 }
613 }
615 /* Hash table of inequalities in a basic map.
616 * "index" is an array of addresses of inequalities in the basic map, some
617 * of which are NULL. The inequalities are hashed on the coefficients
618 * except the constant term.
619 * "size" is the number of elements in the array and is always a power of two
620 * "bits" is the number of bits need to represent an index into the array.
621 * "total" is the total dimension of the basic map.
622 */
628 };
630 /* Fill in the "ci" data structure for holding the inequalities of "bmap".
631 */
634 {
651 }
653 /* Free the memory allocated by create_constraint_index.
654 */
656 {
658 }
660 /* Return the position in ci->index that contains the address of
661 * an inequality that is equal to *ineq up to the constant term,
662 * provided this address is not identical to "ineq".
663 * If there is no such inequality, then return the position where
664 * such an inequality should be inserted.
665 */
667 {
675 }
677 /* Return the position in ci->index that contains the address of
678 * an inequality that is equal to the k'th inequality of "bmap"
679 * up to the constant term, provided it does not point to the very
680 * same inequality.
681 * If there is no such inequality, then return the position where
682 * such an inequality should be inserted.
683 */
686 {
688 }
692 {
694 }
696 /* Fill in the "ci" data structure with the inequalities of "bset".
697 */
700 {
709 }
712 }
714 /* Is the inequality ineq (obviously) redundant with respect
715 * to the constraints in "ci"?
716 *
717 * Look for an inequality in "ci" with the same coefficients and then
718 * check if the contant term of "ineq" is greater than or equal
719 * to the constant term of that inequality. If so, "ineq" is clearly
720 * redundant.
721 *
722 * Note that hash_index_ineq ignores a stored constraint if it has
723 * the same address as the passed inequality. It is ok to pass
724 * the address of a local variable here since it will never be
725 * the same as the address of a constraint in "ci".
726 */
729 {
736 }
738 /* If we can eliminate more than one div, then we need to make
739 * sure we do it from last div to first div, in order not to
740 * change the position of the other divs that still need to
741 * be removed.
742 */
745 {
799 }
801 }
813 }
816 out:
820 }
823 {
835 }
837 }
839 /* Normalize divs that appear in equalities.
840 *
841 * In particular, we assume that bmap contains some equalities
842 * of the form
843 *
844 * a x = m * e_i
845 *
846 * and we want to replace the set of e_i by a minimal set and
847 * such that the new e_i have a canonical representation in terms
848 * of the vector x.
849 * If any of the equalities involves more than one divs, then
850 * we currently simply bail out.
851 *
852 * Let us first additionally assume that all equalities involve
853 * a div. The equalities then express modulo constraints on the
854 * remaining variables and we can use "parameter compression"
855 * to find a minimal set of constraints. The result is a transformation
856 *
857 * x = T(x') = x_0 + G x'
858 *
859 * with G a lower-triangular matrix with all elements below the diagonal
860 * non-negative and smaller than the diagonal element on the same row.
861 * We first normalize x_0 by making the same property hold in the affine
862 * T matrix.
863 * The rows i of G with a 1 on the diagonal do not impose any modulo
864 * constraint and simply express x_i = x'_i.
865 * For each of the remaining rows i, we introduce a div and a corresponding
866 * equality. In particular
867 *
868 * g_ii e_j = x_i - g_i(x')
869 *
870 * where each x'_k is replaced either by x_k (if g_kk = 1) or the
871 * corresponding div (if g_kk != 1).
872 *
873 * If there are any equalities not involving any div, then we
874 * first apply a variable compression on the variables x:
875 *
876 * x = C x'' x'' = C_2 x
877 *
878 * and perform the above parameter compression on A C instead of on A.
879 * The resulting compression is then of the form
880 *
881 * x'' = T(x') = x_0 + G x'
882 *
883 * and in constructing the new divs and the corresponding equalities,
884 * we have to replace each x'', i.e., the x'_k with (g_kk = 1),
885 * by the corresponding row from C_2.
886 */
889 {
898 isl_int v;
930 }
931 }
940 }
946 }
956 }
963 }
968 /* We have to be careful because dropping equalities may reorder them */
978 }
979 }
985 else
986 needed++;
987 }
993 }
1003 else
1011 else
1014 }
1018 }
1025 done:
1029 error:
1035 }
1039 {
1049 }
1051 /* Check whether it is ok to define a div based on an inequality.
1052 * To avoid the introduction of circular definitions of divs, we
1053 * do not allow such a definition if the resulting expression would refer to
1054 * any other undefined divs or if any known div is defined in
1055 * terms of the unknown div.
1056 */
1059 {
1063 /* Not defined in terms of unknown divs */
1071 }
1073 /* No other div defined in terms of this one => avoid loops */
1081 }
1084 }
1086 /* Would an expression for div "div" based on inequality "ineq" of "bmap"
1087 * be a better expression than the current one?
1088 *
1089 * If we do not have any expression yet, then any expression would be better.
1090 * Otherwise we check if the last variable involved in the inequality
1091 * (disregarding the div that it would define) is in an earlier position
1092 * than the last variable involved in the current div expression.
1093 */
1096 {
1113 }
1115 /* Given two constraints "k" and "l" that are opposite to each other,
1116 * except for the constant term, check if we can use them
1117 * to obtain an expression for one of the hitherto unknown divs or
1118 * a "better" expression for a div for which we already have an expression.
1119 * "sum" is the sum of the constant terms of the constraints.
1120 * If this sum is strictly smaller than the coefficient of one
1121 * of the divs, then this pair can be used define the div.
1122 * To avoid the introduction of circular definitions of divs, we
1123 * do not use the pair if the resulting expression would refer to
1124 * any other undefined divs or if any known div is defined in
1125 * terms of the unknown div.
1126 */
1130 {
1135 isl_bool set_div;
1150 else
1155 }
1157 }
1161 {
1165 isl_int sum;
1180 }
1188 }
1203 }
1205 /* We need to break out of the loop after these
1206 * changes since the contents of the hash
1207 * will no longer be valid.
1208 * Plus, we probably we want to regauss first.
1209 */
1217 }
1222 }
1224 /* Detect all pairs of inequalities that form an equality.
1225 *
1226 * isl_basic_map_remove_duplicate_constraints detects at most one such pair.
1227 * Call it repeatedly while it is making progress.
1228 */
1231 {
1243 }
1245 /* Eliminate knowns divs from constraints where they appear with
1246 * a (positive or negative) unit coefficient.
1247 *
1248 * That is, replace
1249 *
1250 * floor(e/m) + f >= 0
1251 *
1252 * by
1253 *
1254 * e + m f >= 0
1255 *
1256 * and
1257 *
1258 * -floor(e/m) + f >= 0
1259 *
1260 * by
1261 *
1262 * -e + m f + m - 1 >= 0
1263 *
1264 * The first conversion is valid because floor(e/m) >= -f is equivalent
1265 * to e/m >= -f because -f is an integral expression.
1266 * The second conversion follows from the fact that
1267 *
1268 * -floor(e/m) = ceil(-e/m) = floor((-e + m - 1)/m)
1269 *
1270 *
1271 * Note that one of the div constraints may have been eliminated
1272 * due to being redundant with respect to the constraint that is
1273 * being modified by this function. The modified constraint may
1274 * no longer imply this div constraint, so we add it back to make
1275 * sure we do not lose any information.
1276 *
1277 * We skip integral divs, i.e., those with denominator 1, as we would
1278 * risk eliminating the div from the div constraints. We do not need
1279 * to handle those divs here anyway since the div constraints will turn
1280 * out to form an equality and this equality can then be used to eliminate
1281 * the div from all constraints.
1282 */
1285 {
1317 else
1327 }
1332 }
1333 }
1336 }
1339 {
1344 isl_bool empty;
1360 /* requires equalities in normal form */
1366 }
1368 }
1371 {
1373 }
1378 {
1410 }
1414 {
1416 }
1419 /* If the only constraints a div d=floor(f/m)
1420 * appears in are its two defining constraints
1421 *
1422 * f - m d >=0
1423 * -(f - (m - 1)) + m d >= 0
1424 *
1425 * then it can safely be removed.
1426 */
1428 {
1437 isl_bool red;
1444 }
1451 }
1454 }
1456 /*
1457 * Remove divs that don't occur in any of the constraints or other divs.
1458 * These can arise when dropping constraints from a basic map or
1459 * when the divs of a basic map have been temporarily aligned
1460 * with the divs of another basic map.
1461 */
1464 {
1471 isl_bool redundant;
1479 }
1481 }
1483 /* Mark "bmap" as final, without checking for obviously redundant
1484 * integer divisions. This function should be used when "bmap"
1485 * is known not to involve any such integer divisions.
1486 */
1489 {
1494 }
1496 /* Mark "bmap" as final, after removing obviously redundant integer divisions.
1497 */
1499 {
1503 }
1506 {
1508 }
1510 /* Remove definition of any div that is defined in terms of the given variable.
1511 * The div itself is not removed. Functions such as
1512 * eliminate_divs_ineq depend on the other divs remaining in place.
1513 */
1516 {
1530 }
1532 }
1534 /* Eliminate the specified variables from the constraints using
1535 * Fourier-Motzkin. The variables themselves are not removed.
1536 */
1539 {
1571 }
1578 n_lower++;
1580 n_upper++;
1581 }
1605 }
1608 }
1620 }
1621 }
1625 error:
1628 }
1632 {
1635 }
1637 /* Eliminate the specified n dimensions starting at first from the
1638 * constraints, without removing the dimensions from the space.
1639 * If the set is rational, the dimensions are eliminated using Fourier-Motzkin.
1640 * Otherwise, they are projected out and the original space is restored.
1641 */
1645 {
1661 }
1668 error:
1671 }
1676 {
1678 }
1680 /* Remove all constraints from "bmap" that reference any unknown local
1681 * variables (directly or indirectly).
1682 *
1683 * Dropping all constraints on a local variable will make it redundant,
1684 * so it will get removed implicitly by
1685 * isl_basic_map_drop_constraints_involving_dims. Some other local
1686 * variables may also end up becoming redundant if they only appear
1687 * in constraints together with the unknown local variable.
1688 * Therefore, start over after calling
1689 * isl_basic_map_drop_constraints_involving_dims.
1690 */
1693 {
1694 isl_bool known;
1719 }
1722 }
1724 /* Remove all constraints from "map" that reference any unknown local
1725 * variables (directly or indirectly).
1726 *
1727 * Since constraints may get dropped from the basic maps,
1728 * they may no longer be disjoint from each other.
1729 */
1732 {
1734 isl_bool known;
1752 }
1758 }
1760 /* Don't assume equalities are in order, because align_divs
1761 * may have changed the order of the divs.
1762 */
1764 {
1777 }
1778 }
1779 }
1783 {
1785 }
1789 {
1803 }
1805 }
1807 }
1811 {
1814 }
1818 {
1828 }
1849 error:
1853 }
1855 /* For each inequality in "ineq" that is a shifted (more relaxed)
1856 * copy of an inequality in "context", mark the corresponding entry
1857 * in "row" with -1.
1858 * If an inequality only has a non-negative constant term, then
1859 * mark it as well.
1860 */
1863 {
1880 isl_bool redundant;
1886 }
1893 }
1896 error:
1899 }
1903 {
1916 isl_bool redundant;
1928 }
1931 error:
1934 }
1936 /* Remove constraints from "bmap" that are identical to constraints
1937 * in "context" or that are more relaxed (greater constant term).
1938 *
1939 * We perform the test for shifted copies on the pure constraints
1940 * in remove_shifted_constraints.
1941 */
1944 {
1953 }
1966 error:
1970 }
1972 /* Does the (linear part of a) constraint "c" involve any of the "len"
1973 * "relevant" dimensions?
1974 */
1976 {
1984 }
1987 }
1989 /* Drop constraints from "bmap" that do not involve any of
1990 * the dimensions marked "relevant".
1991 */
1994 {
2009 }
2016 }
2019 }
2021 /* Update the groups in "group" based on the (linear part of a) constraint "c".
2022 *
2023 * In particular, for any variable involved in the constraint,
2024 * find the actual group id from before and replace the group
2025 * of the corresponding variable by the minimal group of all
2026 * the variables involved in the constraint considered so far
2027 * (if this minimum is smaller) or replace the minimum by this group
2028 * (if the minimum is larger).
2029 *
2030 * At the end, all the variables in "c" will (indirectly) point
2031 * to the minimal of the groups that they referred to originally.
2032 */
2034 {
2051 }
2052 }
2054 /* Allocate an array of groups of variables, one for each variable
2055 * in "context", initialized to zero.
2056 */
2058 {
2065 }
2067 /* Drop constraints from "bmap" that only involve variables that are
2068 * not related to any of the variables marked with a "-1" in "group".
2069 *
2070 * We construct groups of variables that collect variables that
2071 * (indirectly) appear in some common constraint of "bmap".
2072 * Each group is identified by the first variable in the group,
2073 * except for the special group of variables that was already identified
2074 * in the input as -1 (or are related to those variables).
2075 * If group[i] is equal to i (or -1), then the group of i is i (or -1),
2076 * otherwise the group of i is the group of group[i].
2077 *
2078 * We first initialize groups for the remaining variables.
2079 * Then we iterate over the constraints of "bmap" and update the
2080 * group of the variables in the constraint by the smallest group.
2081 * Finally, we resolve indirect references to groups by running over
2082 * the variables.
2083 *
2084 * After computing the groups, we drop constraints that do not involve
2085 * any variables in the -1 group.
2086 */
2089 {
2106 }
2124 }
2126 /* Drop constraints from "context" that are irrelevant for computing
2127 * the gist of "bset".
2128 *
2129 * In particular, drop constraints in variables that are not related
2130 * to any of the variables involved in the constraints of "bset"
2131 * in the sense that there is no sequence of constraints that connects them.
2132 *
2133 * We first mark all variables that appear in "bset" as belonging
2134 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2135 */
2138 {
2159 }
2165 }
2168 }
2170 /* Drop constraints from "context" that are irrelevant for computing
2171 * the gist of the inequalities "ineq".
2172 * Inequalities in "ineq" for which the corresponding element of row
2173 * is set to -1 have already been marked for removal and should be ignored.
2174 *
2175 * In particular, drop constraints in variables that are not related
2176 * to any of the variables involved in "ineq"
2177 * in the sense that there is no sequence of constraints that connects them.
2178 *
2179 * We first mark all variables that appear in "bset" as belonging
2180 * to a "-1" group and then continue with group_and_drop_irrelevant_constraints.
2181 */
2184 {
2205 }
2208 }
2211 }
2213 /* Do all "n" entries of "row" contain a negative value?
2214 */
2216 {
2224 }
2226 /* Update the inequalities in "bset" based on the information in "row"
2227 * and "tab".
2228 *
2229 * In particular, the array "row" contains either -1, meaning that
2230 * the corresponding inequality of "bset" is redundant, or the index
2231 * of an inequality in "tab".
2232 *
2233 * If the row entry is -1, then drop the inequality.
2234 * Otherwise, if the constraint is marked redundant in the tableau,
2235 * then drop the inequality. Similarly, if it is marked as an equality
2236 * in the tableau, then turn the inequality into an equality and
2237 * perform Gaussian elimination.
2238 */
2241 {
2258 }
2268 }
2269 }
2275 }
2277 /* Update the inequalities in "bset" based on the information in "row"
2278 * and "tab" and free all arguments (other than "bset").
2279 */
2284 {
2293 }
2295 /* Remove all information from bset that is redundant in the context
2296 * of context.
2297 * "ineq" contains the (possibly transformed) inequalities of "bset",
2298 * in the same order.
2299 * The (explicit) equalities of "bset" are assumed to have been taken
2300 * into account by the transformation such that only the inequalities
2301 * are relevant.
2302 * "context" is assumed not to be empty.
2303 *
2304 * "row" keeps track of the constraint index of a "bset" inequality in "tab".
2305 * A value of -1 means that the inequality is obviously redundant and may
2306 * not even appear in "tab".
2307 *
2308 * We first mark the inequalities of "bset"
2309 * that are obviously redundant with respect to some inequality in "context".
2310 * Then we remove those constraints from "context" that have become
2311 * irrelevant for computing the gist of "bset".
2312 * Note that this removal of constraints cannot be replaced by
2313 * a factorization because factors in "bset" may still be connected
2314 * to each other through constraints in "context".
2315 *
2316 * If there are any inequalities left, we construct a tableau for
2317 * the context and then add the inequalities of "bset".
2318 * Before adding these inequalities, we freeze all constraints such that
2319 * they won't be considered redundant in terms of the constraints of "bset".
2320 * Then we detect all redundant constraints (among the
2321 * constraints that weren't frozen), first by checking for redundancy in the
2322 * the tableau and then by checking if replacing a constraint by its negation
2323 * would lead to an empty set. This last step is fairly expensive
2324 * and could be optimized by more reuse of the tableau.
2325 * Finally, we update bset according to the results.
2326 */
2329 {
2344 }
2380 }
2404 }
2409 }
2413 error:
2421 }
2423 /* Extract the inequalities of "bset" as an isl_mat.
2424 */
2426 {
2440 }
2442 /* Remove all information from "bset" that is redundant in the context
2443 * of "context", for the case where both "bset" and "context" are
2444 * full-dimensional.
2445 */
2448 {
2453 }
2455 /* Remove all information from "bset" that is redundant in the context
2456 * of "context", for the case where the combined equalities of
2457 * "bset" and "context" allow for a compression that can be obtained
2458 * by preapplication of "T".
2459 *
2460 * "bset" itself is not transformed by "T". Instead, the inequalities
2461 * are extracted from "bset" and those are transformed by "T".
2462 * uset_gist_full then determines which of the transformed inequalities
2463 * are redundant with respect to the transformed "context" and removes
2464 * the corresponding inequalities from "bset".
2465 *
2466 * After preapplying "T" to the inequalities, any common factor is
2467 * removed from the coefficients. If this results in a tightening
2468 * of the constant term, then the same tightening is applied to
2469 * the corresponding untransformed inequality in "bset".
2470 * That is, if after plugging in T, a constraint f(x) >= 0 is of the form
2471 *
2472 * g f'(x) + r >= 0
2473 *
2474 * with 0 <= r < g, then it is equivalent to
2475 *
2476 * f'(x) >= 0
2477 *
2478 * This means that f(x) >= 0 is equivalent to f(x) - r >= 0 in the affine
2479 * subspace compressed by T since the latter would be transformed to
2480 *
2481 * g f'(x) >= 0
2482 */
2486 {
2490 isl_int rem;
2502 }
2525 }
2529 error:
2534 }
2536 /* Project "bset" onto the variables that are involved in "template".
2537 */
2540 {
2549 isl_bool involved;
2558 }
2561 }
2563 /* Remove all information from bset that is redundant in the context
2564 * of context. In particular, equalities that are linear combinations
2565 * of those in context are removed. Then the inequalities that are
2566 * redundant in the context of the equalities and inequalities of
2567 * context are removed.
2568 *
2569 * First of all, we drop those constraints from "context"
2570 * that are irrelevant for computing the gist of "bset".
2571 * Alternatively, we could factorize the intersection of "context" and "bset".
2572 *
2573 * We first compute the intersection of the integer affine hulls
2574 * of "bset" and "context",
2575 * compute the gist inside this intersection and then reduce
2576 * the constraints with respect to the equalities of the context
2577 * that only involve variables already involved in the input.
2578 *
2579 * If two constraints are mutually redundant, then uset_gist_full
2580 * will remove the second of those constraints. We therefore first
2581 * sort the constraints so that constraints not involving existentially
2582 * quantified variables are given precedence over those that do.
2583 * We have to perform this sorting before the variable compression,
2584 * because that may effect the order of the variables.
2585 */
2588 {
2613 }
2618 }
2628 }
2639 }
2642 error:
2646 }
2648 /* Return the number of equality constraints in "bmap" that involve
2649 * local variables. This function assumes that Gaussian elimination
2650 * has been applied to the equality constraints.
2651 */
2653 {
2673 }
2675 /* Construct a basic map in "space" defined by the equality constraints in "eq".
2676 * The constraints are assumed not to involve any local variables.
2677 */
2680 {
2698 }
2703 error:
2708 }
2710 /* Construct and return a variable compression based on the equality
2711 * constraints in "bmap1" and "bmap2" that do not involve the local variables.
2712 * "n1" is the number of (initial) equality constraints in "bmap1"
2713 * that do involve local variables.
2714 * "n2" is the number of (initial) equality constraints in "bmap2"
2715 * that do involve local variables.
2716 * "total" is the total number of other variables.
2717 * This function assumes that Gaussian elimination
2718 * has been applied to the equality constraints in both "bmap1" and "bmap2"
2719 * such that the equality constraints not involving local variables
2720 * are those that start at "n1" or "n2".
2721 *
2722 * If either of "bmap1" and "bmap2" does not have such equality constraints,
2723 * then simply compute the compression based on the equality constraints
2724 * in the other basic map.
2725 * Otherwise, combine the equality constraints from both into a new
2726 * basic map such that Gaussian elimination can be applied to this combination
2727 * and then construct a variable compression from the resulting
2728 * equality constraints.
2729 */
2733 {
2743 }
2748 }
2763 }
2765 /* Extract the stride constraints from "bmap", compressed
2766 * with respect to both the stride constraints in "context" and
2767 * the remaining equality constraints in both "bmap" and "context".
2768 * "bmap_n_eq" is the number of (initial) stride constraints in "bmap".
2769 * "context_n_eq" is the number of (initial) stride constraints in "context".
2770 *
2771 * Let x be all variables in "bmap" (and "context") other than the local
2772 * variables. First compute a variable compression
2773 *
2774 * x = V x'
2775 *
2776 * based on the non-stride equality constraints in "bmap" and "context".
2777 * Consider the stride constraints of "context",
2778 *
2779 * A(x) + B(y) = 0
2780 *
2781 * with y the local variables and plug in the variable compression,
2782 * resulting in
2783 *
2784 * A(V x') + B(y) = 0
2785 *
2786 * Use these constraints to compute a parameter compression on x'
2787 *
2788 * x' = T x''
2789 *
2790 * Now consider the stride constraints of "bmap"
2791 *
2792 * C(x) + D(y) = 0
2793 *
2794 * and plug in x = V*T x''.
2795 * That is, return A = [C*V*T D].
2796 */
2800 {
2829 }
2831 /* Remove the prime factors from *g that have an exponent that
2832 * is strictly smaller than the exponent in "c".
2833 * All exponents in *g are known to be smaller than or equal
2834 * to those in "c".
2835 *
2836 * That is, if *g is equal to
2837 *
2838 * p_1^{e_1} p_2^{e_2} ... p_n^{e_n}
2839 *
2840 * and "c" is equal to
2841 *
2842 * p_1^{f_1} p_2^{f_2} ... p_n^{f_n}
2843 *
2844 * then update *g to
2845 *
2846 * p_1^{e_1 * (e_1 = f_1)} p_2^{e_2 * (e_2 = f_2)} ...
2847 * p_n^{e_n * (e_n = f_n)}
2848 *
2849 * If e_i = f_i, then c / *g does not have any p_i factors and therefore
2850 * neither does the gcd of *g and c / *g.
2851 * If e_i < f_i, then the gcd of *g and c / *g has a positive
2852 * power min(e_i, s_i) of p_i with s_i = f_i - e_i among its factors.
2853 * Dividing *g by this gcd therefore strictly reduces the exponent
2854 * of the prime factors that need to be removed, while leaving the
2855 * other prime factors untouched.
2856 * Repeating this process until gcd(*g, c / *g) = 1 therefore
2857 * removes all undesired factors, without removing any others.
2858 */
2860 {
2861 isl_int t;
2870 }
2872 }
2874 /* Reduce the "n" stride constraints in "bmap" based on a copy "A"
2875 * of the same stride constraints in a compressed space that exploits
2876 * all equalities in the context and the other equalities in "bmap".
2877 *
2878 * If the stride constraints of "bmap" are of the form
2879 *
2880 * C(x) + D(y) = 0
2881 *
2882 * then A is of the form
2883 *
2884 * B(x') + D(y) = 0
2885 *
2886 * If any of these constraints involves only a single local variable y,
2887 * then the constraint appears as
2888 *
2889 * f(x) + m y_i = 0
2890 *
2891 * in "bmap" and as
2892 *
2893 * h(x') + m y_i = 0
2894 *
2895 * in "A".
2896 *
2897 * Let g be the gcd of m and the coefficients of h.
2898 * Then, in particular, g is a divisor of the coefficients of h and
2899 *
2900 * f(x) = h(x')
2901 *
2902 * is known to be a multiple of g.
2903 * If some prime factor in m appears with the same exponent in g,
2904 * then it can be removed from m because f(x) is already known
2905 * to be a multiple of g and therefore in particular of this power
2906 * of the prime factors.
2907 * Prime factors that appear with a smaller exponent in g cannot
2908 * be removed from m.
2909 * Let g' be the divisor of g containing all prime factors that
2910 * appear with the same exponent in m and g, then
2911 *
2912 * f(x) + m y_i = 0
2913 *
2914 * can be replaced by
2915 *
2916 * f(x) + m/g' y_i' = 0
2917 *
2918 * Note that (if g' != 1) this changes the explicit representation
2919 * of y_i to that of y_i', so the integer division at position i
2920 * is marked unknown and later recomputed by a call to
2921 * isl_basic_map_gauss.
2922 */
2925 {
2929 isl_int gcd;
2963 }
2970 error:
2974 }
2976 /* Simplify the stride constraints in "bmap" based on
2977 * the remaining equality constraints in "bmap" and all equality
2978 * constraints in "context".
2979 * Only do this if both "bmap" and "context" have stride constraints.
2980 *
2981 * First extract a copy of the stride constraints in "bmap" in a compressed
2982 * space exploiting all the other equality constraints and then
2983 * use this compressed copy to simplify the original stride constraints.
2984 */
2987 {
3009 }
3011 /* Return a basic map that has the same intersection with "context" as "bmap"
3012 * and that is as "simple" as possible.
3013 *
3014 * The core computation is performed on the pure constraints.
3015 * When we add back the meaning of the integer divisions, we need
3016 * to (re)introduce the div constraints. If we happen to have
3017 * discovered that some of these integer divisions are equal to
3018 * some affine combination of other variables, then these div
3019 * constraints may end up getting simplified in terms of the equalities,
3020 * resulting in extra inequalities on the other variables that
3021 * may have been removed already or that may not even have been
3022 * part of the input. We try and remove those constraints of
3023 * this form that are most obviously redundant with respect to
3024 * the context. We also remove those div constraints that are
3025 * redundant with respect to the other constraints in the result.
3026 *
3027 * The stride constraints among the equality constraints in "bmap" are
3028 * also simplified with respecting to the other equality constraints
3029 * in "bmap" and with respect to all equality constraints in "context".
3030 */
3033 {
3044 }
3050 }
3054 }
3076 }
3095 error:
3099 }
3101 /*
3102 * Assumes context has no implicit divs.
3103 */
3106 {
3117 }
3137 }
3138 }
3142 error:
3146 }
3148 /* Drop all inequalities from "bmap" that also appear in "context".
3149 * "context" is assumed to have only known local variables and
3150 * the initial local variables of "bmap" are assumed to be the same
3151 * as those of "context".
3152 * The constraints of both "bmap" and "context" are assumed
3153 * to have been sorted using isl_basic_map_sort_constraints.
3154 *
3155 * Run through the inequality constraints of "bmap" and "context"
3156 * in sorted order.
3157 * If a constraint of "bmap" involves variables not in "context",
3158 * then it cannot appear in "context".
3159 * If a matching constraint is found, it is removed from "bmap".
3160 */
3163 {
3182 }
3188 }
3192 }
3197 }
3200 }
3203 }
3205 /* Drop all equalities from "bmap" that also appear in "context".
3206 * "context" is assumed to have only known local variables and
3207 * the initial local variables of "bmap" are assumed to be the same
3208 * as those of "context".
3209 *
3210 * Run through the equality constraints of "bmap" and "context"
3211 * in sorted order.
3212 * If a constraint of "bmap" involves variables not in "context",
3213 * then it cannot appear in "context".
3214 * If a matching constraint is found, it is removed from "bmap".
3215 */
3218 {
3242 }
3246 }
3251 }
3254 }
3257 }
3259 /* Remove the constraints in "context" from "bmap".
3260 * "context" is assumed to have explicit representations
3261 * for all local variables.
3262 *
3263 * First align the divs of "bmap" to those of "context" and
3264 * sort the constraints. Then drop all constraints from "bmap"
3265 * that appear in "context".
3266 */
3269 {
3284 }
3303 error:
3307 }
3309 /* Replace "map" by the disjunct at position "pos" and free "context".
3310 */
3313 {
3320 }
3322 /* Remove the constraints in "context" from "map".
3323 * If any of the disjuncts in the result turns out to be the universe,
3324 * then return this universe.
3325 * "context" is assumed to have explicit representations
3326 * for all local variables.
3327 */
3330 {
3340 }
3359 }
3366 error:
3370 }
3372 /* Remove the constraints in "context" from "set".
3373 * If any of the disjuncts in the result turns out to be the universe,
3374 * then return this universe.
3375 * "context" is assumed to have explicit representations
3376 * for all local variables.
3377 */
3380 {
3383 }
3385 /* Remove the constraints in "context" from "map".
3386 * If any of the disjuncts in the result turns out to be the universe,
3387 * then return this universe.
3388 * "context" is assumed to consist of a single disjunct and
3389 * to have explicit representations for all local variables.
3390 */
3393 {
3398 }
3400 /* Replace "map" by a universe map in the same space and free "drop".
3401 */
3404 {
3411 }
3413 /* Return a map that has the same intersection with "context" as "map"
3414 * and that is as "simple" as possible.
3415 *
3416 * If "map" is already the universe, then we cannot make it any simpler.
3417 * Similarly, if "context" is the universe, then we cannot exploit it
3418 * to simplify "map"
3419 * If "map" and "context" are identical to each other, then we can
3420 * return the corresponding universe.
3421 *
3422 * If either "map" or "context" consists of multiple disjuncts,
3423 * then check if "context" happens to be a subset of "map",
3424 * in which case all constraints can be removed.
3425 * In case of multiple disjuncts, the standard procedure
3426 * may not be able to detect that all constraints can be removed.
3427 *
3428 * If none of these cases apply, we have to work a bit harder.
3429 * During this computation, we make use of a single disjunct context,
3430 * so if the original context consists of more than one disjunct
3431 * then we need to approximate the context by a single disjunct set.
3432 * Simply taking the simple hull may drop constraints that are
3433 * only implicitly available in each disjunct. We therefore also
3434 * look for constraints among those defining "map" that are valid
3435 * for the context. These can then be used to simplify away
3436 * the corresponding constraints in "map".
3437 */
3440 {
3444 isl_bool subset;
3455 }
3471 }
3487 list);
3488 }
3490 error:
3494 }
3498 {
3500 }
3504 {
3507 }
3511 {
3514 }
3518 {
3523 }
3527 {
3529 }
3531 /* Compute the gist of "bmap" with respect to the constraints "context"
3532 * on the domain.
3533 */
3536 {
3542 }
3546 {
3550 }
3554 {
3558 }
3562 {
3566 }
3570 {
3572 }
3574 /* Quick check to see if two basic maps are disjoint.
3575 * In particular, we reduce the equalities and inequalities of
3576 * one basic map in the context of the equalities of the other
3577 * basic map and check if we get a contradiction.
3578 */
3581 {
3613 }
3621 }
3630 }
3634 disjoint:
3638 error:
3642 }
3646 {
3649 }
3651 /* Does "test" hold for all pairs of basic maps in "map1" and "map2"?
3652 */
3656 {
3667 }
3668 }
3671 }
3673 /* Are "map1" and "map2" obviously disjoint, based on information
3674 * that can be derived without looking at the individual basic maps?
3675 *
3676 * In particular, if one of them is empty or if they live in different spaces
3677 * (ignoring parameters), then they are clearly disjoint.
3678 */
3681 {
3682 isl_bool disjoint;
3683 isl_bool match;
3707 }
3709 /* Are "map1" and "map2" obviously disjoint?
3710 *
3711 * If one of them is empty or if they live in different spaces (ignoring
3712 * parameters), then they are clearly disjoint.
3713 * This is checked by isl_map_plain_is_disjoint_global.
3714 *
3715 * If they have different parameters, then we skip any further tests.
3716 *
3717 * If they are obviously equal, but not obviously empty, then we will
3718 * not be able to detect if they are disjoint.
3719 *
3720 * Otherwise we check if each basic map in "map1" is obviously disjoint
3721 * from each basic map in "map2".
3722 */
3725 {
3726 isl_bool disjoint;
3727 isl_bool intersect;
3728 isl_bool match;
3743 }
3745 /* Are "map1" and "map2" disjoint?
3746 * The parameters are assumed to have been aligned.
3747 *
3748 * In particular, check whether all pairs of basic maps are disjoint.
3749 */
3752 {
3754 }
3756 /* Are "map1" and "map2" disjoint?
3757 *
3758 * They are disjoint if they are "obviously disjoint" or if one of them
3759 * is empty. Otherwise, they are not disjoint if one of them is universal.
3760 * If the two inputs are (obviously) equal and not empty, then they are
3761 * not disjoint.
3762 * If none of these cases apply, then check if all pairs of basic maps
3763 * are disjoint after aligning the parameters.
3764 */
3766 {
3767 isl_bool disjoint;
3768 isl_bool intersect;
3796 }
3798 /* Are "bmap1" and "bmap2" disjoint?
3799 *
3800 * They are disjoint if they are "obviously disjoint" or if one of them
3801 * is empty. Otherwise, they are not disjoint if one of them is universal.
3802 * If none of these cases apply, we compute the intersection and see if
3803 * the result is empty.
3804 */
3807 {
3808 isl_bool disjoint;
3809 isl_bool intersect;
3838 }
3840 /* Are "bset1" and "bset2" disjoint?
3841 */
3844 {
3846 }
3850 {
3852 }
3854 /* Are "set1" and "set2" disjoint?
3855 */
3857 {
3859 }
3861 /* Is "v" equal to 0, 1 or -1?
3862 */
3864 {
3866 }
3868 /* Check if we can combine a given div with lower bound l and upper
3869 * bound u with some other div and if so return that other div.
3870 * Otherwise return -1.
3871 *
3872 * We first check that
3873 * - the bounds are opposites of each other (except for the constant
3874 * term)
3875 * - the bounds do not reference any other div
3876 * - no div is defined in terms of this div
3877 *
3878 * Let m be the size of the range allowed on the div by the bounds.
3879 * That is, the bounds are of the form
3880 *
3881 * e <= a <= e + m - 1
3882 *
3883 * with e some expression in the other variables.
3884 * We look for another div b such that no third div is defined in terms
3885 * of this second div b and such that in any constraint that contains
3886 * a (except for the given lower and upper bound), also contains b
3887 * with a coefficient that is m times that of b.
3888 * That is, all constraints (except for the lower and upper bound)
3889 * are of the form
3890 *
3891 * e + f (a + m b) >= 0
3892 *
3893 * Furthermore, in the constraints that only contain b, the coefficient
3894 * of b should be equal to 1 or -1.
3895 * If so, we return b so that "a + m b" can be replaced by
3896 * a single div "c = a + m b".
3897 */
3900 {
3922 }
3932 }
3944 }
3955 }
3968 }
3973 }
3977 }
3979 /* Internal data structure used during the construction and/or evaluation of
3980 * an inequality that ensures that a pair of bounds always allows
3981 * for an integer value.
3982 *
3983 * "tab" is the tableau in which the inequality is evaluated. It may
3984 * be NULL until it is actually needed.
3985 * "v" contains the inequality coefficients.
3986 * "g", "fl" and "fu" are temporary scalars used during the construction and
3987 * evaluation.
3988 */
3992 isl_int g;
3993 isl_int fl;
3994 isl_int fu;
3995 };
3997 /* Free all the memory allocated by the fields of "data".
3998 */
4000 {
4006 }
4008 /* Is the inequality stored in data->v satisfied by "bmap"?
4009 * That is, does it only attain non-negative values?
4010 * data->tab is a tableau corresponding to "bmap".
4011 */
4014 {
4025 }
4027 /* Given a lower and an upper bound on div i, do they always allow
4028 * for an integer value of the given div?
4029 * Determine this property by constructing an inequality
4030 * such that the property is guaranteed when the inequality is nonnegative.
4031 * The lower bound is inequality l, while the upper bound is inequality u.
4032 * The constructed inequality is stored in data->v.
4033 *
4034 * Let the upper bound be
4035 *
4036 * -n_u a + e_u >= 0
4037 *
4038 * and the lower bound
4039 *
4040 * n_l a + e_l >= 0
4041 *
4042 * Let n_u = f_u g and n_l = f_l g, with g = gcd(n_u, n_l).
4043 * We have
4044 *
4045 * - f_u e_l <= f_u f_l g a <= f_l e_u
4046 *
4047 * Since all variables are integer valued, this is equivalent to
4048 *
4049 * - f_u e_l - (f_u - 1) <= f_u f_l g a <= f_l e_u + (f_l - 1)
4050 *
4051 * If this interval is at least f_u f_l g, then it contains at least
4052 * one integer value for a.
4053 * That is, the test constraint is
4054 *
4055 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 >= f_u f_l g
4056 *
4057 * or
4058 *
4059 * f_l e_u + f_u e_l + f_l - 1 + f_u - 1 + 1 - f_u f_l g >= 0
4060 *
4061 * If the coefficients of f_l e_u + f_u e_l have a common divisor g',
4062 * then the constraint can be scaled down by a factor g',
4063 * with the constant term replaced by
4064 * floor((f_l e_{u,0} + f_u e_{l,0} + f_l - 1 + f_u - 1 + 1 - f_u f_l g)/g').
4065 * Note that the result of applying Fourier-Motzkin to this pair
4066 * of constraints is
4067 *
4068 * f_l e_u + f_u e_l >= 0
4069 *
4070 * If the constant term of the scaled down version of this constraint,
4071 * i.e., floor((f_l e_{u,0} + f_u e_{l,0})/g') is equal to the constant
4072 * term of the scaled down test constraint, then the test constraint
4073 * is known to hold and no explicit evaluation is required.
4074 * This is essentially the Omega test.
4075 *
4076 * If the test constraint consists of only a constant term, then
4077 * it is sufficient to look at the sign of this constant term.
4078 */
4081 {
4106 }
4116 }
4118 /* Remove more kinds of divs that are not strictly needed.
4119 * In particular, if all pairs of lower and upper bounds on a div
4120 * are such that they allow at least one integer value of the div,
4121 * then we can eliminate the div using Fourier-Motzkin without
4122 * introducing any spurious solutions.
4123 *
4124 * If at least one of the two constraints has a unit coefficient for the div,
4125 * then the presence of such a value is guaranteed so there is no need to check.
4126 * In particular, the value attained by the bound with unit coefficient
4127 * can serve as this intermediate value.
4128 */
4131 {
4154 isl_bool has_int;
4162 }
4183 }
4186 }
4190 }
4194 }
4197 }
4208 error:
4213 }
4215 /* Given a pair of divs div1 and div2 such that, except for the lower bound l
4216 * and the upper bound u, div1 always occurs together with div2 in the form
4217 * (div1 + m div2), where m is the constant range on the variable div1
4218 * allowed by l and u, replace the pair div1 and div2 by a single
4219 * div that is equal to div1 + m div2.
4220 *
4221 * The new div will appear in the location that contains div2.
4222 * We need to modify all constraints that contain
4223 * div2 = (div - div1) / m
4224 * The coefficient of div2 is known to be equal to 1 or -1.
4225 * (If a constraint does not contain div2, it will also not contain div1.)
4226 * If the constraint also contains div1, then we know they appear
4227 * as f (div1 + m div2) and we can simply replace (div1 + m div2) by div,
4228 * i.e., the coefficient of div is f.
4229 *
4230 * Otherwise, we first need to introduce div1 into the constraint.
4231 * Let l be
4232 *
4233 * div1 + f >=0
4234 *
4235 * and u
4236 *
4237 * -div1 + f' >= 0
4238 *
4239 * A lower bound on div2
4240 *
4241 * div2 + t >= 0
4242 *
4243 * can be replaced by
4244 *
4245 * m div2 + div1 + m t + f >= 0
4246 *
4247 * An upper bound
4248 *
4249 * -div2 + t >= 0
4250 *
4251 * can be replaced by
4252 *
4253 * -(m div2 + div1) + m t + f' >= 0
4254 *
4255 * These constraint are those that we would obtain from eliminating
4256 * div1 using Fourier-Motzkin.
4257 *
4258 * After all constraints have been modified, we drop the lower and upper
4259 * bound and then drop div1.
4260 * Since the new div is only placed in the same location that used
4261 * to store div2, but otherwise has a different meaning, any possible
4262 * explicit representation of the original div2 is removed.
4263 */
4266 {
4268 isl_int m;
4290 else
4293 }
4297 }
4306 }
4310 }
4312 /* First check if we can coalesce any pair of divs and
4313 * then continue with dropping more redundant divs.
4314 *
4315 * We loop over all pairs of lower and upper bounds on a div
4316 * with coefficient 1 and -1, respectively, check if there
4317 * is any other div "c" with which we can coalesce the div
4318 * and if so, perform the coalescing.
4319 */
4322 {
4345 }
4346 }
4347 }
4352 }
4355 }
4357 /* Are the "n" coefficients starting at "first" of inequality constraints
4358 * "i" and "j" of "bmap" equal to each other?
4359 */
4362 {
4364 }
4366 /* Are the "n" coefficients starting at "first" of inequality constraints
4367 * "i" and "j" of "bmap" opposite to each other?
4368 */
4371 {
4373 }
4375 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4376 * apart from the constant term?
4377 */
4379 {
4384 }
4386 /* Are inequality constraints "i" and "j" of "bmap" equal to each other,
4387 * apart from the constant term and the coefficient at position "pos"?
4388 */
4391 {
4397 }
4399 /* Are inequality constraints "i" and "j" of "bmap" opposite to each other,
4400 * apart from the constant term and the coefficient at position "pos"?
4401 */
4404 {
4410 }
4412 /* Restart isl_basic_map_drop_redundant_divs after "bmap" has
4413 * been modified, simplying it if "simplify" is set.
4414 * Free the temporary data structure "pairs" that was associated
4415 * to the old version of "bmap".
4416 */
4419 {
4424 }
4426 /* Is "div" the single unknown existentially quantified variable
4427 * in inequality constraint "ineq" of "bmap"?
4428 * "div" is known to have a non-zero coefficient in "ineq".
4429 */
4432 {
4435 isl_bool known;
4445 isl_bool known;
4454 }
4457 }
4459 /* Does integer division "div" have coefficient 1 in inequality constraint
4460 * "ineq" of "map"?
4461 */
4463 {
4471 }
4473 /* Turn inequality constraint "ineq" of "bmap" into an equality and
4474 * then try and drop redundant divs again,
4475 * freeing the temporary data structure "pairs" that was associated
4476 * to the old version of "bmap".
4477 */
4480 {
4484 }
4486 /* Drop the integer division at position "div", along with the two
4487 * inequality constraints "ineq1" and "ineq2" in which it appears
4488 * from "bmap" and then try and drop redundant divs again,
4489 * freeing the temporary data structure "pairs" that was associated
4490 * to the old version of "bmap".
4491 */
4495 {
4502 }
4505 }
4507 /* Given two inequality constraints
4508 *
4509 * f(x) + n d + c >= 0, (ineq)
4510 *
4511 * with d the variable at position "pos", and
4512 *
4513 * f(x) + c0 >= 0, (lower)
4514 *
4515 * compute the maximal value of the lower bound ceil((-f(x) - c)/n)
4516 * determined by the first constraint.
4517 * That is, store
4518 *
4519 * ceil((c0 - c)/n)
4520 *
4521 * in *l.
4522 */
4525 {
4529 }
4531 /* Given two inequality constraints
4532 *
4533 * f(x) + n d + c >= 0, (ineq)
4534 *
4535 * with d the variable at position "pos", and
4536 *
4537 * -f(x) - c0 >= 0, (upper)
4538 *
4539 * compute the minimal value of the lower bound ceil((-f(x) - c)/n)
4540 * determined by the first constraint.
4541 * That is, store
4542 *
4543 * ceil((-c1 - c)/n)
4544 *
4545 * in *u.
4546 */
4549 {
4553 }
4555 /* Given a lower bound constraint "ineq" on "div" in "bmap",
4556 * does the corresponding lower bound have a fixed value in "bmap"?
4557 *
4558 * In particular, "ineq" is of the form
4559 *
4560 * f(x) + n d + c >= 0
4561 *
4562 * with n > 0, c the constant term and
4563 * d the existentially quantified variable "div".
4564 * That is, the lower bound is
4565 *
4566 * ceil((-f(x) - c)/n)
4567 *
4568 * Look for a pair of constraints
4569 *
4570 * f(x) + c0 >= 0
4571 * -f(x) + c1 >= 0
4572 *
4573 * i.e., -c1 <= -f(x) <= c0, that fix ceil((-f(x) - c)/n) to a constant value.
4574 * That is, check that
4575 *
4576 * ceil((-c1 - c)/n) = ceil((c0 - c)/n)
4577 *
4578 * If so, return the index of inequality f(x) + c0 >= 0.
4579 * Otherwise, return -1.
4580 */
4582 {
4599 }
4603 }
4604 }
4621 }
4623 /* Given a lower bound constraint "ineq" on the existentially quantified
4624 * variable "div", such that the corresponding lower bound has
4625 * a fixed value in "bmap", assign this fixed value to the variable and
4626 * then try and drop redundant divs again,
4627 * freeing the temporary data structure "pairs" that was associated
4628 * to the old version of "bmap".
4629 * "lower" determines the constant value for the lower bound.
4630 *
4631 * In particular, "ineq" is of the form
4632 *
4633 * f(x) + n d + c >= 0,
4634 *
4635 * while "lower" is of the form
4636 *
4637 * f(x) + c0 >= 0
4638 *
4639 * The lower bound is ceil((-f(x) - c)/n) and its constant value
4640 * is ceil((c0 - c)/n).
4641 */
4644 {
4645 isl_int c;
4658 }
4660 /* Remove divs that are not strictly needed based on the inequality
4661 * constraints.
4662 * In particular, if a div only occurs positively (or negatively)
4663 * in constraints, then it can simply be dropped.
4664 * Also, if a div occurs in only two constraints and if moreover
4665 * those two constraints are opposite to each other, except for the constant
4666 * term and if the sum of the constant terms is such that for any value
4667 * of the other values, there is always at least one integer value of the
4668 * div, i.e., if one plus this sum is greater than or equal to
4669 * the (absolute value) of the coefficient of the div in the constraints,
4670 * then we can also simply drop the div.
4671 *
4672 * If an existentially quantified variable does not have an explicit
4673 * representation, appears in only a single lower bound that does not
4674 * involve any other such existentially quantified variables and appears
4675 * in this lower bound with coefficient 1,
4676 * then fix the variable to the value of the lower bound. That is,
4677 * turn the inequality into an equality.
4678 * If for any value of the other variables, there is any value
4679 * for the existentially quantified variable satisfying the constraints,
4680 * then this lower bound also satisfies the constraints.
4681 * It is therefore safe to pick this lower bound.
4682 *
4683 * The same reasoning holds even if the coefficient is not one.
4684 * However, fixing the variable to the value of the lower bound may
4685 * in general introduce an extra integer division, in which case
4686 * it may be better to pick another value.
4687 * If this integer division has a known constant value, then plugging
4688 * in this constant value removes the existentially quantified variable
4689 * completely. In particular, if the lower bound is of the form
4690 * ceil((-f(x) - c)/n) and there are two constraints, f(x) + c0 >= 0 and
4691 * -f(x) + c1 >= 0 such that ceil((-c1 - c)/n) = ceil((c0 - c)/n),
4692 * then the existentially quantified variable can be assigned this
4693 * shared value.
4694 *
4695 * We skip divs that appear in equalities or in the definition of other divs.
4696 * Divs that appear in the definition of other divs usually occur in at least
4697 * 4 constraints, but the constraints may have been simplified.
4698 *
4699 * If any divs are left after these simple checks then we move on
4700 * to more complicated cases in drop_more_redundant_divs.
4701 */
4704 {
4744 }
4748 }
4749 }
4757 }
4760 else
4780 pairs);
4784 pairs);
4786 }
4803 else
4810 }
4813 }
4820 error:
4824 }
4826 /* Consider the coefficients at "c" as a row vector and replace
4827 * them with their product with "T". "T" is assumed to be a square matrix.
4828 */
4830 {
4852 }
4854 /* Plug in T for the variables in "bmap" starting at "pos".
4855 * T is a linear unimodular matrix, i.e., without constant term.
4856 */
4859 {
4888 }
4892 error:
4896 }
4898 /* Remove divs that are not strictly needed.
4899 *
4900 * First look for an equality constraint involving two or more
4901 * existentially quantified variables without an explicit
4902 * representation. Replace the combination that appears
4903 * in the equality constraint by a single existentially quantified
4904 * variable such that the equality can be used to derive
4905 * an explicit representation for the variable.
4906 * If there are no more such equality constraints, then continue
4907 * with isl_basic_map_drop_redundant_divs_ineq.
4908 *
4909 * In particular, if the equality constraint is of the form
4910 *
4911 * f(x) + \sum_i c_i a_i = 0
4912 *
4913 * with a_i existentially quantified variable without explicit
4914 * representation, then apply a transformation on the existentially
4915 * quantified variables to turn the constraint into
4916 *
4917 * f(x) + g a_1' = 0
4918 *
4919 * with g the gcd of the c_i.
4920 * In order to easily identify which existentially quantified variables
4921 * have a complete explicit representation, i.e., without being defined
4922 * in terms of other existentially quantified variables without
4923 * an explicit representation, the existentially quantified variables
4924 * are first sorted.
4925 *
4926 * The variable transformation is computed by extending the row
4927 * [c_1/g ... c_n/g] to a unimodular matrix, obtaining the transformation
4928 *
4929 * [a_1'] [c_1/g ... c_n/g] [ a_1 ]
4930 * [a_2'] [ a_2 ]
4931 * ... = U ....
4932 * [a_n'] [ a_n ]
4933 *
4934 * with [c_1/g ... c_n/g] representing the first row of U.
4935 * The inverse of U is then plugged into the original constraints.
4936 * The call to isl_basic_map_simplify makes sure the explicit
4937 * representation for a_1' is extracted from the equality constraint.
4938 */
4941 {
4976 }
4995 }
4997 /* Does "bmap" satisfy any equality that involves more than 2 variables
4998 * and/or has coefficients different from -1 and 1?
4999 */
5001 {
5030 }
5033 }
5035 /* Remove any common factor g from the constraint coefficients in "v".
5036 * The constant term is stored in the first position and is replaced
5037 * by floor(c/g). If any common factor is removed and if this results
5038 * in a tightening of the constraint, then set *tightened.
5039 */
5042 {
5062 }
5064 /* If "bmap" is an integer set that satisfies any equality involving
5065 * more than 2 variables and/or has coefficients different from -1 and 1,
5066 * then use variable compression to reduce the coefficients by removing
5067 * any (hidden) common factor.
5068 * In particular, apply the variable compression to each constraint,
5069 * factor out any common factor in the non-constant coefficients and
5070 * then apply the inverse of the compression.
5071 * At the end, we mark the basic map as having reduced constants.
5072 * If this flag is still set on the next invocation of this function,
5073 * then we skip the computation.
5074 *
5075 * Removing a common factor may result in a tightening of some of
5076 * the constraints. If this happens, then we may end up with two
5077 * opposite inequalities that can be replaced by an equality.
5078 * We therefore call isl_basic_map_detect_inequality_pairs,
5079 * which checks for such pairs of inequalities as well as eliminate_divs_eq
5080 * and isl_basic_map_gauss if such a pair was found.
5081 *
5082 * Note that this function may leave the result in an inconsistent state.
5083 * In particular, the constraints may not be gaussed.
5084 * Unfortunately, isl_map_coalesce actually depends on this inconsistent state
5085 * for some of the test cases to pass successfully.
5086 * Any potential modification of the representation is therefore only
5087 * performed on a single copy of the basic map.
5088 */
5091 {
5125 }
5140 }
5155 }
5156 }
5159 error:
5164 }
5166 /* Shift the integer division at position "div" of "bmap"
5167 * by "shift" times the variable at position "pos".
5168 * "pos" is as determined by isl_basic_map_offset, i.e., pos == 0
5169 * corresponds to the constant term.
5170 *
5171 * That is, if the integer division has the form
5172 *
5173 * floor(f(x)/d)
5174 *
5175 * then replace it by
5176 *
5177 * floor((f(x) + shift * d * x_pos)/d) - shift * x_pos
5178 */
5181 {
5200 }
5206 }
5214 }
5217 }